Dielectrics: An Atomic View — Question 7

Question 7

A measured polarization must respect the molecular alignment bound

A sample contains number density n of permanent dipoles of fixed magnitude p. Induced polarization is negligible. A reported polarization along the applied field is P.

1. Express the mean alignment and derive its allowed range.

2. For $n=1.00\times10^{27}$ m$^{-3}$ and $p=3.00\times10^{-30}$ C m, assess reports P=.00180 C/m$^2$ and P=.00360 C/m$^2$.

Question 7 - Solution

A measured polarization must respect the molecular alignment bound

(a) Method and calculation

$$P=np\langle\cos\theta\rangle\Rightarrow\langle\cos\theta\rangle=P/(np).$$$$|\cos\theta|\le1\Rightarrow|P|\le np.$$

(b) Interpretation and further calculation

$$np=.00300\,\mathrm{C/m^2};\quad .00180/.00300=.600\quad\text{admissible}.$$$$.00360/.00300=1.200>1\quad\text{incompatible with the stated model}.$$

Physical check

A report beyond saturation requires changed assumptions or measurement correction; increasing field cannot align a fixed dipole beyond cosine one.